Sequences Involving Own Elements <=m

[Leroy Quet]

Here are a couple interesting sequences which are not in the EIS, for the 
most part. (But I did not submit them to Superseeker.)

1)

a(1) = 1;

a(m+1) = a(m) + (largest element of {a} <= m)


1, 2, 4, 6, 10, 14, 20, 26, 32, 38, 48,...

(sort of a recursive analog to the trianglular numbers)

Is there a closed-form for this sequence?

Also, a(m+1) = a(m) + a(k(m)), 
where k(m) is the number of elements of {a} which are <= m.

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2)

b(1) = 1; 

b(m+1) = b(m) * k(m),

where k(m) is the number of elements of {b(j)}, 1<=j<=m, which are <= m.

1, 1, 2, 6, 18, 54, 216,...

What would be the asymptotics of this sequence?

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3)

c(0) = 1;

c(m) = m + (largest element of {c} <= m);

1, 2, 4, 5, 8, 10, 11, 12, 16, 17, 20, 22, 24, 25, 26, 27, 32,...

Is this sequence A037988 + 1 ?

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4)

If we take sequence A095114:

http://www.research.att.com/projects/OEIS?Anum=A095114


But we generalize by instead having the first term = n, n = positive 
integer,

we then have this array, where d(n,m+1) = 
d(n,m) + (number of elements of d(n,k),1<=k<=m, which are <= m):

d(n,m)

1, 2, 4, 6, 9, 12, 16, 20, 24,...
2, 2, 4, 6, 9, 12, 16, 20, 24,...
3, 3, 3, 6, 9, 12, 16, 20, 24,...
4, 4, 4, 4, 8, 12, 16, 20, 25,...<- (note 25, not 24)
5, 5, 5, 5, 5, 10, 15, 20, 25,...
6, 6, 6, 6, 6, 6, 12, 18, 24,...

If the sequence is growing faster than that of another n, it will 
eventually slow more, then rise faster as a result of slowing, etc, I 
think.

Anything else which can be said about this, or any of the above, 
sequences?

thanks,
Leroy Quet